A BEAUTIFUL Theorem on when MONOIDS are GROUPS! | Abstract Algebra Axioms!

A monoid is a set with an associative binary operation that has an identity. A monoid is a group if every element has a two-sided inverse. In this video, we will prove that if every element of a monoid has a one-sided inverse (which could vary between being a left-inverse or a right-inverse depending on the element), then the monoid is a group! In other words, the one-sided inverse of each element is actually a two-sided inverse.

The video is designed for people studying introductory abstract algebra/group theory, who have a passing familiarity with the group axioms. We revisit the inverse axiom in a group since it is important to the proof. We apply simple creative tricks with the group axioms in order to establish the statement.

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